The W. Thurston Algorithm Applied to Real Polynomial Maps

Entropy stability and Milnor-Thurston invariants for Bowen-Series-like maps

We define a family of discontinuous maps on the circle, called Bowen-Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor-Thurston theory for one dimensional maps.

An algebraic characterization of expanding Thurston maps

Let $f: S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.

Weak expansion properties and large deviation principles for expanding Thurston maps

In this paper, we prove that an expanding Thurston map $f\colon S^2 \rightarrow S^2$ is asymptotically $h$-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is $h$-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on $S^2$, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the …

临界不动的反Thurston映射的分类

Classification of critically fixed anti-Thurston maps

该研究建立了Thurston理论的反向定向版本与反Thurston映射的典范分解，完成了临界不动反Thurston映射的完整组合分类，并给出了相关应用。

Thurston equivalence for rational maps with clusters

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number $ρ$ and the critical displacement $δ$ of the cluster cycle. The same result will also be proved in the case that the rational map is quadratic and has a period two cluster cycle, but that the statement is no longer true in the higher degree case.

The growth rate inequality for Thurston maps with non hyperbolic orbifolds

Let $f: S^2 \to S^2$ be a continuous map of degree $d$, $|d|>1$, and let $N_nf$ denote the number of fixed points of $f^n$. We show that if $f$ is a Thurston map with non hyperbolic orbifold, then either the growth rate inequality $\limsup \frac{1}{n} \log N_nf\geq \log |d|$ holds for $f$ or $f$ has exactly two critical points which are fixed and totally invariant.

Integrality and Thurston Rigidity for Bicritical PCF Polynomials

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.

Rationality is decidable for nearly Euclidean Thurston maps

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at most $C$, the problem of determining whether $F$ is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of $C$ and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality.

临界固定图示映射：分类、识别与扭转

Critically fixed Thurston maps: classification, recognition, and twisting

本文研究了临界固定图示映射的分类、识别及扭转问题，通过吹胀操作将图与同胚映射联系起来，提供了分类方法和重建算法。

余弦 Thurston 映射的部分刻画

A Partial Characterization of Cosine Thurston Maps

本文引入余弦 Thurston 映射，利用 Hubbard 等人的技术证明了具有严格预周期临界点的后奇异有限拓扑余弦映射在无退化 Levy 循环时组合等价于唯一的 $C_λ(z) = λ\cos z$。

On thurston's core entropy algorithm dedicated to the memory of tan lei

The core entropy of polynomials, recently introduced by W. Thurston, is a dy-namical invariant extending topological entropy for real maps to complex polynomials, whence providing a new tool to study the parameter space of polynomials. The base is a combinatorial algorithm allowing for the computation of the core entropy given by Thurston, but without supplying a proof. In this paper, we will describe his algorithm and prove its validity.

Matings of Cubic polynomials with a fixed critical point, Part I: Thurston Obstructions

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the mating $f \mate g$ contains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.

On Net Maps: Examples and Nonexistence Results

A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. Nearly Euclidean Thurston (NET) maps are simple generalizations of rational Lattes maps. We investigate when such a map has the property that the associated pullback map on Teichmuller space is constant. We also show that no Thurston map of degree 2 has constant pullback map

Thurston's algorithm and rational maps from quadratic polynomial matings

Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the Riemann sphere. Given a pair of polynomials of the form $z^2+c$ that are postcritically finite, there is a fast test on the constant parameters to determine whether this map $F$ exists---but this test is not constructive. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, $F$. Thi…

具有小后临界集的有限与无限度Thurston映射

Finite and infinite degree Thurston maps with a small postsingular set

本文研究拓扑球面上除单个本性奇点外处处有定义的Thurston映射，针对具有四个后临界值的广泛情形建立了W. Thurston刻画定理的类似物，并通过分析一维Teichmüller空间上的拉回映射推导出相应Hurwitz类的性质。

Thurston type Theorem for sub-hyperbolic rational maps

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem.

扩张Thurston映射的基态与周期轨道

Ground states and periodic orbits for expanding Thurston maps

本文研究扩张Thurston映射的最大化测度与基态，证明典型周期优化猜想成立，并推广到Misiurewicz–Thurston有理映射。

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed `Herman map' is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result gives an answer to a problem of McMullen in a sense and enables us to prove a Thurston-type theorem for …

Quadratic Thurston maps with few postcritical points

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.

带额外标记点的有限次与无限次Thurston映射

Finite and infinite degree Thurston maps with extra marked points

该研究针对带额外标记点的Thurston映射族，证明了后奇点有限全纯实现的存在等价于无退化Levy闭轨，并通过分析拉回映射的不变复子流形，结合复动力学与双曲几何工具验证了结论。

Algorithmic aspects of branched coverings IV/V. Expanding maps

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as maps with "contracting biset". We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Ree…

Prime orbit theorems for expanding Thurston maps

We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity assumption. More precisely, by studying dynamical zeta functions and, more generally, dynamical Dirichlet series for $f$, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function $ϕ$ on $S^2$ satisfying some additional regularity conditions,…

具有 Julia 集 $\mathbb{S}^2$ 的扩展 Thurston 映射的拆配

Unmating of expanding Thurston maps with Julia set $\mathbb{S}^2$

本文针对临界预周期扩展 Thurston 映射，通过调整拆配算法以适应同伦意义下完全不变的非 Jordan 曲线，实现了无需迭代即可进行多项式拆配。

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $μ_ϕ$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued Hölder continuous potential $ϕ$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,μ_ϕ)$ is exact, and in particular, mixing and ergodic. Moreover, we establish ve…

Levy and Thurston obstructions of finite subdivision rules

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and…

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $γ\colon S^1\to S^2$ (onto), such that $F\circ γ(z) = γ(z^d)$ (for all $z\in S^1$).

扩张Thurston映射子系统的热力学形式II

Thermodynamic formalism for subsystems of expanding Thurston maps II

本文研究扩张Thurston映射的强素子系统的热力学形式，证明了实值Hölder连续势下平衡态的唯一性与遍历性质，建立原像等分布性，并得到无周期临界点情形下的2级大偏差原理。

在收缩区间内计数素轨道的扩张图映射

Counting prime orbits in shrinking intervals for expanding Thurston maps

本文研究了扩张图映射中原始周期轨道在收缩区间内的计数问题，通过局部中心极限定理和渐近估计，揭示了非均匀扩展动力学下的细粒度行为。

An algebraic approach to certain cases of Thurston rigidity

In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's rigidity theorem is that the curves C_{1,n} and C_{2,m} intersect transversally. We give a purely algebraic proof that the intersection points are 3-adically integral and use this to prove transversality. We also prove an analogous result when c_1 or c_2 or both are taken to be preperiodic with tail length exactly 1.

Anosov叶面的Cannon--Thurston映射

Cannon--Thurston maps for Anosov foliations

研究Anosov叶面的universal circles，证明其存在Cannon--Thurston型映射，揭示流形基本群的伪Anosov动力学。

Thurston equivalence of topological polynomials

We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.

Algebraic stability of meromorphic maps descended from Thurston's pullback maps

Let $ϕ:S^2 \to S^2$ be an orientation-preserving branched covering whose post-critical set has finite cardinality $n$. If $ϕ$ has a fully ramified periodic point $p_{\infty}$ and satisfies certain additional conditions, then, by work of Koch, $ϕ$ induces a meromorphic self-map $R_ϕ$ on the moduli space $\mathcal{M}_{0,n}$; $R_ϕ$ descends from Thurston's pullback map on Teichmüller space. Here, we relate the dynamics of $R_ϕ$ on $\mathcal{M}_{0,n}$ to the dynamics of $ϕ$ on $S^2$. Let $\ell$ be the length of the periodic cycle in which the fully ramified point $p_{\infty}$ lies; we show that $R…

扩张 Thurston 映射子系统的热力学形式与大偏差渐近

Thermodynamic formalism for subsystems of expanding Thurston maps and large deviations asymptotics

本文受 Sullivan 字典启发引入扩张 Thurston 映射的子系统，利用热力学形式在视觉度量下证明强不可约子系统的变分原理与平衡态存在性，并由此建立大偏差渐近。

图林型映射的刚性和准等距统一化

Rigidity and quasisymmetric uniformization of Thurston-type maps

本文研究图林型映射的刚性与准等距统一化，证明了无不变线场猜想并建立了相关定理。

Pullback invariants of Thurston maps

Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in $S^2- P$, a linear operator on the free $\R$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties.

The core entropy for polynomials of higher degree

As defined by W. Thurston, the core entropy of a polynomial is the entropy of the restriction to its Hubbard tree. For each d >= 2, we study the core entropy as a function on the parameter space of polynomials of degree d, and prove it varies continuously both as a function of the combinatorial data and of the coefficients of the polynomials. This generalizes a conjecture of Thurston for quadratic polynomials.

An Algebraic Proof of Thurston's Rigidity for a Polynomial

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic proof of this fact for tamely ramified maps for which at least one of the critical points is periodic. We also produce wildly ramified counterexamples.

Invariant spanning trees for quadratic rational maps

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes - the ivy graph.

Thurston maps and asymptotic upper curvature

A Thurston map is a branched covering map from $§^2$ to $§^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on $§^2$ containing $\post(f)$. The boundary at infinity of $\G$ with associated visual metrics can be identified with $§^2$ equipped with the visual metric induced by the expanding Thurston map $f$. We define asymptotic upper curvature of an expanding Thurston map $f$ to be the asymptotic upper curvature of the associated Gromov hyperbolic graph, and establish a conn…

Topological entropy of pseudo-Anosov maps from a typical Thurston's construction

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss …

Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with…

Expansion properties for finite subdivision rules II

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.

Expanding Thurston maps as quotients

A Thurston map is a branched covering map $f\colon S^2\to S^2$ that is postcritically finite. Mating of polynomials, introduced by Douady and Hubbard, is a method to geometrically combine the Julia sets of two polynomials (and their dynamics) to form a rational map. We show that for every expanding Thurston map $f$ every sufficiently high iterate $F=f^n$ is obtained as the mating of two polynomials. One obtains a concise description of $F$ via critical portraits. The proof is based on the construction of the invariant Peano curve from Meyer. As another consequence we obtain a large number of f…

Continuity of core entropy of quadratic polynomials

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps: the real segment is replaced by an invariant tree, known as Hubbard tree, which lives inside the filled Julia set. We prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.

扩张 Thurston 映射的素轨道定理：Lattès 映射与分裂 Ruelle 算子

Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators

本文为二维球面上的扩张 Thurston 映射建立了素数定理的类比，证明在满足 α-强不可积性条件的权重下，原始周期轨道数渐近服从对数积分并带指数误差界，结果覆盖后临界有限有理映射，并对 Lattès 映射给出更强结论。

The Shape of Thurston's Master Teapot

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$. We show that the Thurston set described above is not equal to the Thurston set for postcritically finite tent maps, and we provide an arithmetic explanation for why certain gaps appear in plots of finite approximations of the Thurston set.

二次Misiurewicz-Thurston多项式的因式分解

Factorization of the quadratic Misiurewicz-Thurston polynomials

本文给出了二次Misiurewicz-Thurston多项式的完整因式分解，将其根分类为双曲点和前周期点，并确定了各自的重数。

膨胀 Thurston 映射的素轨道定理：Dirichlet 级数与 orbifold

Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds

本文利用权重、Dirichlet 级数与 orbifold 相关框架，为膨胀 Thurston 映射建立本原周期轨道的素数型渐近定理，并推广至 Julia 集覆盖整个球面的临界后有限有理映射。

后奇异有限整函数的动力学逼近

Dynamical approximations of postsingularly finite entire maps

本文证明后奇异有限整函数可由后临界有限复多项式序列逼近且保持后奇异动力学共轭，并通过引入整 Thurston 映射的组合收敛概念，证明其诱导的拉回映射在 Teichmüller 空间上局部一致收敛。

上同调分形、Cannon-Thurston映射与测地流

Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

本文通过光线追踪生成双曲三维流形的上同调分形图像，证明其与Cannon-Thurston映射的对应关系，并揭示其极限为无穷远球面上的分布。

The Thurston Algorithm for quadratic matings

Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless the orbifold of f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of p…

无限维 Thurston 理论与超越动力学 II：具有逃逸奇异轨道的整函数的分类

Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits

本文利用无限维 Teichmüller 空间上的迭代程序，将 Thurston 对有理函数的经典拓扑刻画推广至无限标记点情形，从而分类所有奇异值沿互不相交射线逃逸的多项式与指数复合超越整函数。

扩张 Thurston 映射的素轨道定理：强不可积条件的通有性

Prime orbit theorems for expanding Thurston maps: Genericity of strong non-integrability condition

本文证明在扩张 Thurston 映射的素轨道定理中，α-强不可积条件在 α-Hölder 连续函数类中具有通有性，从而完善了该系列关于素数定理类比的研究。

A disconnected deformation space of rational maps

Let $f:(\mathbb{P}^1,P)\to(\mathbb{P}^1,P)$ be a postcritically finite rational map with postcritical set $P$. William Thurston showed that $f$ induces a holomorphic pullback map $σ_f:\mathcal{T}_P\to\mathcal{T}_P$ on the Teichmüller space ${\mathcal T}_P:=\mathrm{Teich}(\mathbb{P}^1,P)$. If $f$ is not a flexible Lattès map, Thurston proved that $σ_f$ has a unique fixed point. In his PhD thesis, Adam Epstein generalized Thurston's ideas and defined a deformation space associated to a rational map $f:(\mathbb{P}^1,A)\to (\mathbb{P}^1,B)$ where $A \subseteq B$, allowing for maps $f$ which are no…